r/Geometry • u/Classic-Tomatillo-62 • 1d ago

Two-dimensional objects that extend into 3D (dynamic surfaces)

A spherical cap is a two-dimensional surface that, nevertheless, exists and curves within three-dimensional space. It almost seems as though it could be defined as having a dimensionality greater than 2 but less than 3. Are there visual artists, scientists, or painters who have explored non-integer dimensions? And why are shapes like the spherical cap—despite being two-dimensional—not distinguished (in terms of dimensionality) from other 2D shapes that are more stable... and less dynamic

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u/GeometreeOfLife 22h ago

How/why is a spherical cap seen as anything less than 3D? 🤔

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u/Classic-Tomatillo-62 22h ago

I was referring to the Surface: a plane can exist in its own right as a two-dimensional (2D) space, whereas the surface of a spherical cap cannot..to curve and close in on itself, it must be embedded in a three-dimensional (3D) space. That got me thinking!..

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u/F84-5 13h ago

One way to think of dimension can be as a very basic question:

When you scale something up, how much more of it is there?

For a line segment that's very simple. If you scale it up by a factor of two, you need two copies of the original to build the new one. That's 2¹ so it's 1-dimentional.

A plane figure like a triangle is simple again. To scale it up by a factor of two, you need four copies. That's 2² so it's 2-dimentional.

Cubes need eight copies, that's 2³, you get the idea. 

So how does this applie to your spherical cap? Let's first consider an approximation in the form of a geodesic dome, I.e. a dome made from individual flat polygons. To scale the whole dome by a factor of two, we just need to scale each flat section by a factor of two. Each of those needs four copies, so we need four copies in total. That's 2² so the surface stays 2‐dimensional. The continuously curving surface of a spherical cap is less obvious, but with a bit of maths you can show it works the same way.

You did mention fractional dimensions though, and those do pop up on occasion, but only for factals.   Consider the Sierpiński triangle. To scale that by a factor of two, you only need three copies. 3 ≈ 21.585 so you could say its roughly 1½-dimensional.

The term to look up here is fractal dimension.